The following is a report containing the models fit to data from 2019-11-29 to 2021-01-01. Data are measured in XXX Millions. This report summarizes results for Net Foreign Assets.
The RMSE is defined as
\[RMSE=\sqrt{\sum_{t\in\mathcal{T_{eval}}}(y_t-\hat{y}_t(h))^2}\] where \(y_t\) is the observed value and \(\hat{y}_t(h)\) is the h-step ahead forecast of \(y_t\). The RMSE for each model and HORIZON is shown below. Lower values indicate better forecasts, the best individual model is in bold, while combinations are colored.
The MAE is defined as
\[MAE=\sum_{t\in\mathcal{T_{eval}}} \left| y_t-\hat{y}_t(h) \right|\] where \(y_t\) is the observed value and \(\hat{y}_t(h)\) is the h-step ahead forecast of \(y_t\). The RMSE for each model and HORIZON is shown below. Lower values indicate better forecasts.
To evaluate the accuracy of a 95% prediction interval, one can consider the proportion of times the observed value is outside the prediction interval. This is summarised in the table below. Values close to 0.05 indicate the best forecasts with values below 0.05 indicating conservative prediction intervals.
Net Foreign Assets over the training period is plotted below. Missing values (mostly Fridays and Saturdays) are linearly interpolated.
While the mean of Net Foreign Assets follows a random walk, the first difference does exhibit conditional heteroskedasticity.
Since the fitted values of a GARCH model are simple equal to the (constant) sample mean, the data are plotted with 95% intervals. This shows how each model captures time varying volatility.
A exponentially weighted moving average (EWMA) model is a special case of the GARCH model with a single parameter. The estimated parameters are summarised below:
——————————— * GARCH Model Fit ———————————*
GARCH Model : iGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm
Estimate Std. Error t value Pr(>|t|)
mu -0.000672 0.014425 -0.046621 0.96281 omega 0.000000 NA NA NA alpha1 0.032530 0.003880 8.383309 0.00000 beta1 0.967470 NA NA NA
Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu -0.000672 0.018183 -0.036984 0.970498 omega 0.000000 NA NA NA alpha1 0.032530 0.010252 3.172936 0.001509 beta1 0.967470 NA NA NA
LogLikelihood : -5004.499
Akaike 2.7425 Bayes 2.7459 Shibata 2.7425 Hannan-Quinn 2.7437
statistic p-value
Lag[1] 416.6 0 Lag[2*(p+q)+(p+q)-1][2] 419.6 0 Lag[4*(p+q)+(p+q)-1][5] 438.7 0 d.o.f=0 H0 : No serial correlation
statistic p-value
Lag[1] 208.0 0 Lag[2*(p+q)+(p+q)-1][5] 218.7 0 Lag[4*(p+q)+(p+q)-1][9] 239.7 0 d.o.f=2
Statistic Shape Scale P-Value
ARCH Lag[3] 7.765 0.500 2.000 5.327e-03 ARCH Lag[5] 20.116 1.440 1.667 2.003e-05 ARCH Lag[7] 39.415 2.315 1.543 5.090e-10
Joint Statistic: 0.4042 Individual Statistics:
mu 0.004732 alpha1 0.402164
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 0.61 0.749 1.07 Individual Statistic: 0.35 0.47 0.75
t-value prob sig
Sign Bias 1.726 8.436e-02 Negative Sign Bias 6.101 1.167e-09 Positive Sign Bias 10.777 1.114e-26 Joint Effect 153.722 4.147e-33 **
group statistic p-value(g-1) 1 20 574.1 1.489e-109 2 30 636.8 1.491e-115 3 40 712.2 1.342e-124 4 50 726.0 2.696e-121
Elapsed time : 0.05102801
An standard GARCH(1,1) is regularly used to model conditional heteroskedasticty. The estimated parameters are summarised below:
——————————— * GARCH Model Fit ———————————*
GARCH Model : sGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm
Estimate Std. Error t value Pr(>|t|)
mu 0.001849 0.012821 0.14421 0.88533 omega 0.571130 0.026051 21.92335 0.00000 alpha1 0.488887 0.039417 12.40285 0.00000 beta1 0.013750 0.021109 0.65136 0.51481
Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.001849 0.009785 0.18895 0.85013 omega 0.571130 0.058849 9.70496 0.00000 alpha1 0.488887 0.073568 6.64536 0.00000 beta1 0.013750 0.040633 0.33840 0.73507
LogLikelihood : -4875.447
Akaike 2.6729 Bayes 2.6797 Shibata 2.6729 Hannan-Quinn 2.6754
statistic p-value
Lag[1] 159.2 0 Lag[2*(p+q)+(p+q)-1][2] 170.3 0 Lag[4*(p+q)+(p+q)-1][5] 200.4 0 d.o.f=0 H0 : No serial correlation
statistic p-value
Lag[1] 0.3028 5.821e-01 Lag[2*(p+q)+(p+q)-1][5] 1.2214 8.079e-01 Lag[4*(p+q)+(p+q)-1][9] 21.6888 7.334e-05 d.o.f=2
Statistic Shape Scale P-Value
ARCH Lag[3] 0.06307 0.500 2.000 8.017e-01 ARCH Lag[5] 1.42629 1.440 1.667 6.119e-01 ARCH Lag[7] 26.27941 2.315 1.543 1.526e-06
Joint Statistic: 5.1086 Individual Statistics:
mu 0.09198 omega 3.79540 alpha1 0.54163 beta1 1.14985
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.07 1.24 1.6 Individual Statistic: 0.35 0.47 0.75
t-value prob sig
Sign Bias 0.04015 0.9680
Negative Sign Bias 0.95582 0.3392
Positive Sign Bias 0.30883 0.7575
Joint Effect 1.67421 0.6427
group statistic p-value(g-1) 1 20 510.0 4.518e-96 2 30 549.9 1.525e-97 3 40 554.0 3.041e-92 4 50 578.7 1.306e-91
Elapsed time : 0.2722399
An eGARCH(1,1) extends the GARCH model by allowings for asymmetric effects. The estimated parameters are summarised below:
——————————— * GARCH Model Fit ———————————*
GARCH Model : eGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm
Estimate Std. Error t value Pr(>|t|)
mu 0.012058 0.013732 0.87808 0.379901 omega -0.041826 0.024244 -1.72523 0.084486 alpha1 0.052851 0.024117 2.19144 0.028420 beta1 0.288775 0.045214 6.38683 0.000000 gamma1 0.655781 0.034563 18.97359 0.000000
Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.012058 0.013780 0.87504 0.381552 omega -0.041826 0.052070 -0.80327 0.421821 alpha1 0.052851 0.035173 1.50261 0.132939 beta1 0.288775 0.087746 3.29102 0.000998 gamma1 0.655781 0.053105 12.34883 0.000000
LogLikelihood : -4862.87
Akaike 2.6666 Bayes 2.6751 Shibata 2.6666 Hannan-Quinn 2.6696
statistic p-value
Lag[1] 162.8 0 Lag[2*(p+q)+(p+q)-1][2] 170.2 0 Lag[4*(p+q)+(p+q)-1][5] 198.3 0 d.o.f=0 H0 : No serial correlation
statistic p-value
Lag[1] 0.532 4.658e-01 Lag[2*(p+q)+(p+q)-1][5] 1.847 6.551e-01 Lag[4*(p+q)+(p+q)-1][9] 23.291 2.786e-05 d.o.f=2
Statistic Shape Scale P-Value
ARCH Lag[3] 1.243 0.500 2.000 2.650e-01 ARCH Lag[5] 2.408 1.440 1.667 3.880e-01 ARCH Lag[7] 28.515 2.315 1.543 3.966e-07
Joint Statistic: 5.2232 Individual Statistics:
mu 0.08241 omega 3.74179 alpha1 0.21411 beta1 2.60814 gamma1 2.83494
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.28 1.47 1.88 Individual Statistic: 0.35 0.47 0.75
t-value prob sig
Sign Bias 0.2011 0.8407
Negative Sign Bias 0.1403 0.8884
Positive Sign Bias 0.3311 0.7406
Joint Effect 0.2891 0.9621
group statistic p-value(g-1) 1 20 530.0 2.833e-100 2 30 555.5 1.052e-98 3 40 578.7 2.890e-97 4 50 637.1 2.589e-103
Elapsed time : 0.4344161
A GJR GARCH(1,1) extends the GARCH model by allowings for asymmetric effects. The estimated parameters are summarised below:
——————————— * GARCH Model Fit ———————————*
GARCH Model : gjrGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm
Estimate Std. Error t value Pr(>|t|)
mu 0.018459 0.014204 1.29949 0.193775 omega 0.574773 0.027025 21.26814 0.000000 alpha1 0.585565 0.059663 9.81446 0.000000 beta1 0.014462 0.022947 0.63023 0.528541 gamma1 -0.209572 0.075691 -2.76879 0.005626
Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.018459 0.011244 1.64171 0.100651 omega 0.574773 0.057909 9.92540 0.000000 alpha1 0.585565 0.112390 5.21012 0.000000 beta1 0.014462 0.041721 0.34664 0.728864 gamma1 -0.209572 0.123409 -1.69819 0.089472
LogLikelihood : -4871.608
Akaike 2.6714 Bayes 2.6799 Shibata 2.6714 Hannan-Quinn 2.6744
statistic p-value
Lag[1] 160.3 0 Lag[2*(p+q)+(p+q)-1][2] 171.0 0 Lag[4*(p+q)+(p+q)-1][5] 200.8 0 d.o.f=0 H0 : No serial correlation
statistic p-value
Lag[1] 0.3346 0.5629747 Lag[2*(p+q)+(p+q)-1][5] 1.2607 0.7984087 Lag[4*(p+q)+(p+q)-1][9] 21.0558 0.0001072 d.o.f=2
Statistic Shape Scale P-Value
ARCH Lag[3] 0.00388 0.500 2.000 9.503e-01 ARCH Lag[5] 1.56501 1.440 1.667 5.754e-01 ARCH Lag[7] 25.57290 2.315 1.543 2.332e-06
Joint Statistic: 5.1464 Individual Statistics:
mu 0.08485 omega 3.83598 alpha1 0.54184 beta1 1.39182 gamma1 0.32691
Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.28 1.47 1.88 Individual Statistic: 0.35 0.47 0.75
t-value prob sig
Sign Bias 0.2526 0.8006
Negative Sign Bias 0.2376 0.8122
Positive Sign Bias 0.3511 0.7255
Joint Effect 0.2419 0.9706
group statistic p-value(g-1) 1 20 514.2 5.941e-97 2 30 532.8 5.012e-94 3 40 570.2 1.512e-95 4 50 604.6 8.826e-97
Elapsed time : 0.5956001